## Fabrication of mesoscale polymeric templates for three-dimensional disordered photonic materials |

Optics Express, Vol. 21, Issue 1, pp. 1057-1065 (2013)

http://dx.doi.org/10.1364/OE.21.001057

Acrobat PDF (2540 KB)

### Abstract

We report on the mesoscale fabrication and characterization of polymeric templates for isotropic photonic materials derived from hyper-uniform point patterns using direct laser writing in a polymer photoresist. We study experimentally the microscopic structure by electron microscopy and small angle light scattering. Reducing the refractive index mismatch by liquid infiltration we find good agreement between the scattering data and numerical calculations in the single scattering limit. Our work thus demonstrates the feasibility of fabricating such random designer materials on technologically relevant length scales.

© 2013 OSA

## 1. Introduction

**k**. For a periodic one-dimensional structure with a lattice constant

*a*forbidden wave vectors appear around the edge of the Brillouin zone

*k*=

*π*/

*a*, the latter being a signature of back-reflection from Bragg planes with scattering vectors

*q*= 2

*k*= 2

*π*/

*a*[1]. Despite the beauty of Blochs theory it is well known that the model covers only a small part of actual electron transport phenomena in metals or semiconductors. A prime example is the fact that many alloys and even liquids display metallic or semiconducting properties [2

2. W. H. Wang, C. Dong, and C. H. Shek, “Bulk metallic glasses,” Mat Sci Eng R **44**, 45–89 (2004). [CrossRef]

3. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys Rev Lett **58**, 2059–2062 (1987). [CrossRef] [PubMed]

4. S. John, “Localization of photons in certain disordered dielectric superlattices,” Phys Rev Lett **58**, 2486–2489 (1987). [CrossRef] [PubMed]

5. C. Lopez, “Materials aspects of photonic crystals,” Adv. Mater. **15**, 1679–1704 (2003). [CrossRef]

6. M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nature Materials **3**, 444–447 (2004). [CrossRef] [PubMed]

7. M. Florescu, S. Torquato, and P. J. Steinhardt, “Designer disordered materials with large, complete photonic band gaps,” Proc. Natl. Acad. Sci. **106**, 20658–20663 (2009). [CrossRef] [PubMed]

8. S. F. Liew, J.-K. Yang, H. Noh, C. F. Schreck, E. R. Dufresne, C. S. OHern, and H. Cao, “Photonic band gaps in three-dimensional network structures with short-range order,” Phys. Rev. A. **84**, 063818 (2011). [CrossRef]

9. M. Florescu, S. Torquato, and P. J. Steinhardt, “Complete band gaps in two-dimensional photonic quasicrystals,” Phys. Rev. B **80**, 155112 (2009). [CrossRef]

10. W. N. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, “Experimental measurement of the photonic properties of icosahedral quasicrystals,” Nature **436**, 993–996 (2005). [CrossRef] [PubMed]

12. C. J. Jin, X. D. Meng, B. Y. Cheng, Z. L. Li, and D. Z. Zhang, “Photonic gap in amorphous photonic materials,” Phys. Rev. B. **63**, 195107 (2001). [CrossRef]

13. M. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and S. Mordechai, “Amorphous photonic lattices: band gaps, effective mass, and suppressed transport,” Phys. Rev. Lett. **106**, 193904 (2011) [CrossRef] [PubMed]

14. L. F. Rojas-Ochoa, J. M. Mendez-Alcaraz, P. Schurtenberger, J. J. Saenz, and F. Scheffold, “Photonic properties of strongly correlated colloidal liquids,” Phys. Rev. Lett. **93**, 073903 (2004). [CrossRef] [PubMed]

15. M. Reufer, L. F. Rojas-Ochoa, P. Schurtenberger, J. J. Saenz, and F. Scheffold, “Transport of light in amorphous photonic materials,” Appl. Phys. Lett. **91**, 171904 (2007). [CrossRef]

16. J. F. Galisteo-Lopez, M. Ibisate, R. Sapienza, L. S. Froufe-Prez, A. Blanco, and C. Lopez, “Self-assembled photonic structures,” Adv Mater **23**, 30–69 (2011). [CrossRef]

17. H. Noh, J.-K. Yang, S. F. Liew, M. J. Rooks, G. S. Solomon, and H. Cao, “Control of lasing in biomimetic structures with short-range order,” Phys Rev Lett **106**, 183901 (2011). [CrossRef] [PubMed]

4. S. John, “Localization of photons in certain disordered dielectric superlattices,” Phys Rev Lett **58**, 2486–2489 (1987). [CrossRef] [PubMed]

7. M. Florescu, S. Torquato, and P. J. Steinhardt, “Designer disordered materials with large, complete photonic band gaps,” Proc. Natl. Acad. Sci. **106**, 20658–20663 (2009). [CrossRef] [PubMed]

## 2. Results

### 2.1. Nanofabrication of designer disordered materials in a polymer photoresist

^{2}. When writing thinner rods the structure becomes mechanically unstable while thicker rods lead to overfilling of the structure. The aspect ratio of ca. 3.0 of the rods is dictated by the point spread function of the illuminating microscope objective during the writing process, which is set by the refractive index of about 1.52 of the material and the numerical aperture of 1.4. For the parameters chosen we estimate the volume-filling fraction of the rods to be roughly 8 %.

6. M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nature Materials **3**, 444–447 (2004). [CrossRef] [PubMed]

^{2}cross section in our case) and not by a layer-by-layer process, commonly employed in lower-resolution commercial 3D printing techniques used for rapid prototyping or manufacturing. Therefore, in our fabrication process particular care must be taken how to set up the writing protocol since there exist no obvious rules how to write a random free-standing network structure at optimal resolution. Moreover, it must be ensured that the structure remains mechanically stable in a soft gel photoresist throughout the writing process of about 1 h. This task is further complicated by intrinsic mechanical stresses created upon exposure of the photoresist. The latter leads to substantial deformations if the written lines are not attached within seconds to a mechanically rigid superstructure. To overcome these challenging problems, encountered in our initial fabrication attempts, we have developed an optimized writing protocol. We first divide the entire volume in cubic sub-volumes of side length roughly 1.5

*D*(where

*D*is the average distance between nearest points in the underlying point pattern). We sort the rods in such a way that i) once all rods belonging to a certain cube are written, we proceed to a neighboring cube, ii) first all cubes closest to the substrate are filled with rods, then the ones lying higher above and so on until the whole network has been written. Figure 2 displays a representative set of electron micrographs of fabricated structures. We succeeded to write structures with either a square (65 × 65

*μ*m

^{2}) or circular (diameter 65

*μ*m) footprint with heights varying between

*h*= 4 – 12

*μ*m. Structures higher than 4

*μ*m were surrounded by a massive wall (Fig. 2) for enhanced mechanical stability [6

6. M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nature Materials **3**, 444–447 (2004). [CrossRef] [PubMed]

### 2.2. Structural analysis by light scattering

*λ*= 632.8 nm), a focusing lens (focal length

*f*= 50 mm), two diaphragms to suppress stray light and a white screen, positioned at a distance of

*z*= 125mm from the sample, with a central absorbing beam block. The scattered light pattern is photographed off the white screen using a digital camera. The setup has been calibrated using a small pinhole. Histogram normalization has been applied to all images displayed in Fig. 3. This procedure represents a linear transformation of an image where the value

*P*of each pixel is scaled according to [24]:

_{in}*P*= 255(

_{out}*P*−

_{in}*c*)/(

*d*−

*c*). Here

*c*and

*d*are the

*x*-th and (100 –

*x*)th percentile in the histogram of pixel intensity values. All values

*P*smaller than zero are set to 0, while those larger than 255 are set to 255. In our case

_{out}*x*= 0.1. Before calculating radial averages of experimental diffraction patterns, several data processing steps were performed. Despite the use of diaphragms we could still observe some stray light contributing to the image. This contribution has to be subtracted from the raw data. To this end we have acquired an image from a bare glass substrate (empty cell) inserted into the laser beam. This image is subsequently subtracted from the raw data. The modulus of the scattering vector

**q**is calculated from the radial distance

*u*from the center via the relation tan(

*θ*) =

*u/z*and

*q*= (4

*π*/

*λ*)sin(

*θ*/2). These relations apply also for the (toluene or toluene/chlorobenzene) infiltrated structures since for small angles the reduction of the wavelength

*λ*/

*n*within the sample is to a good approximation offset by refraction at the flat sample-air interface. Another small correction results from the actual detector acceptance angle being different for each scattering angle

*θ*. This gives rise to a correction [23

23. F. Ferri, “Use of a charge coupled device camera for low-angle elastic light scattering,” Rev. Sci. Instrum. **68**, 2265–2274 (1997). [CrossRef]

*θ*)

^{−3}.

*l*≃ 3

_{s}*μ*m, short even compared to the lowest sample studied with

*h*= 4

*μ*m. In order to reduce scattering we infiltrate the sample first with isopropanol (

*n*= 1.377) and then with toluol (

*n*= 1.496), which leads to a gradual reduction of multiple scattering and a sharpening of the diffraction ring, Fig. 3(b), 3(c), while at the same time the ring position remains unchanged. For the latter case the refractive index of the polymeric structure is almost matched and the direct beam is attenuated only by a few percent signaling the absence of multiple scattering. Similar results are obtained for the

*h*= 8

*μ*m structures when using a 1:2 mixture of toluene/chlorobenzene (

*n*= 1.517) as an index matching fluid (Fig. 4(b). In both cases a clear and pronounced ring appears in the scattering pattern at

*q*∼ 2

*π*/

*d*and for smaller

*q*-values scattering is strongly suppressed. Spatial speckle fluctuations can be largely (although not entirely) suppressed by taking radial averages as shown in Fig. 4.

### 2.3. Comparison with numerical calculations

25. B.T. Draine and P.J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A **1**, 1491–1499 (1994). [CrossRef]

*I*(

**q**) scattered by the structure for a scattering wave vector

**q**=

**k**−

**k**, where

_{0}**k**denotes the scattered wave vector and

**k**the incident wave vector. Lines plotted in Fig. 4 have been obtained by averaging over several realizations of a structure of a given height

_{0}*h*. As shown in Fig. 3(d) the numerical results reproduce both the ring diffraction as well as the superimposed random specular structure. A comparison of the radially averaged numerical data with experiment, Fig. 4, reveals a very good match with no adjustable parameters except for the absolute scale of intensities.

## 3. Discussion

*h*= 4

*μ*m to

*h*= 8

*μ*m. Preliminary numerical results (data not shown) indicate that the peak height saturates for heights larger than 20

*μ*m. Moreover both the experimental and numerical results show that the extrapolated values for

*S*(

*q*→ 0) are finite. Due to experimental difficulties accessing wavenumbers close to the primary beam we are unable to clearly distinguish finite-size effects from residual contaminations and experimental artifacts. Similarly the accuracy of our numerical calculations for small wavenumbers is limited due to finite size effects. Nevertheless, since the seed structure is hyperuniform and since the quality of our polymeric templates is very high, we do believe that the polymeric template possesses the contemplated structural properties and thus should likely give rise to a full PBG when transferred into high-index dielectric.

*n*∼ 3.6 and a volume filling fraction of about 20% [8

8. S. F. Liew, J.-K. Yang, H. Noh, C. F. Schreck, E. R. Dufresne, C. S. OHern, and H. Cao, “Photonic band gaps in three-dimensional network structures with short-range order,” Phys. Rev. A. **84**, 063818 (2011). [CrossRef]

*n*= 3.6 for infrared wavelengths) by double inversion retaining the original topology [26

26. A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener, and G. von Freymann, “Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths,” Nat. Materials **5**, 942–945 (2006). [CrossRef]

27. I. Staude, M. Thiel, S. Essig, C. Wolff, K. Busch, G. von Freymann, and M. Wegener, “Fabrication and characterization of silicon woodpile photonic crystals with a complete bandgap at telecom wavelengths,” Opt. Lett. **35**, 1094–1096 (2010). [CrossRef] [PubMed]

*λ*∼ 4

*π*/

*q*∼ 6

_{max}*μ*m, about four times larger than typical telecommunication wavelengths of

*λ*∼ 1.5

*μ*m [8

8. S. F. Liew, J.-K. Yang, H. Noh, C. F. Schreck, E. R. Dufresne, C. S. OHern, and H. Cao, “Photonic band gaps in three-dimensional network structures with short-range order,” Phys. Rev. A. **84**, 063818 (2011). [CrossRef]

## 4. Summary and conclusion

## Acknowledgments

## References and links

1. | J. D. Joannopoulos, |

2. | W. H. Wang, C. Dong, and C. H. Shek, “Bulk metallic glasses,” Mat Sci Eng R |

3. | E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys Rev Lett |

4. | S. John, “Localization of photons in certain disordered dielectric superlattices,” Phys Rev Lett |

5. | C. Lopez, “Materials aspects of photonic crystals,” Adv. Mater. |

6. | M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nature Materials |

7. | M. Florescu, S. Torquato, and P. J. Steinhardt, “Designer disordered materials with large, complete photonic band gaps,” Proc. Natl. Acad. Sci. |

8. | S. F. Liew, J.-K. Yang, H. Noh, C. F. Schreck, E. R. Dufresne, C. S. OHern, and H. Cao, “Photonic band gaps in three-dimensional network structures with short-range order,” Phys. Rev. A. |

9. | M. Florescu, S. Torquato, and P. J. Steinhardt, “Complete band gaps in two-dimensional photonic quasicrystals,” Phys. Rev. B |

10. | W. N. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, “Experimental measurement of the photonic properties of icosahedral quasicrystals,” Nature |

11. | P. J. Steinhardt and D. P. Divincenzo, |

12. | C. J. Jin, X. D. Meng, B. Y. Cheng, Z. L. Li, and D. Z. Zhang, “Photonic gap in amorphous photonic materials,” Phys. Rev. B. |

13. | M. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and S. Mordechai, “Amorphous photonic lattices: band gaps, effective mass, and suppressed transport,” Phys. Rev. Lett. |

14. | L. F. Rojas-Ochoa, J. M. Mendez-Alcaraz, P. Schurtenberger, J. J. Saenz, and F. Scheffold, “Photonic properties of strongly correlated colloidal liquids,” Phys. Rev. Lett. |

15. | M. Reufer, L. F. Rojas-Ochoa, P. Schurtenberger, J. J. Saenz, and F. Scheffold, “Transport of light in amorphous photonic materials,” Appl. Phys. Lett. |

16. | J. F. Galisteo-Lopez, M. Ibisate, R. Sapienza, L. S. Froufe-Prez, A. Blanco, and C. Lopez, “Self-assembled photonic structures,” Adv Mater |

17. | H. Noh, J.-K. Yang, S. F. Liew, M. J. Rooks, G. S. Solomon, and H. Cao, “Control of lasing in biomimetic structures with short-range order,” Phys Rev Lett |

18. | C. E. Zachary, Y. Jiao, and S. Torquato, “Hyperuniform long-range correlations are a signature of disordered jammed hard-particle packings,” Phys. Rev. Lett. |

19. | W. N. Man, M. Florescu, K. Matsuyama, P. Yadak, S. Torquato, P. J. Steinhardt, and P. Chaikin, “Experimental observation of photonic bandgaps in hyperuniform disordered material” in Proceedings of the Conference on Lasers and Electro-Optics (Cleo) and Quantum Electronics and Laser Science Conference (Qels) (2010). |

20. | C. Song, P. Wang, and H. A. Makse, “A phase diagram for jammed matter,” Nature |

21. | A. Donev, S. Torquato, and F. H. Stillinger, “Pair correlation function characteristics of nearly jammed disordered and ordered hard-sphere packings,” Phys. Rev. E |

22. | J.-L. Barrat and J.-P. Hansen, |

23. | F. Ferri, “Use of a charge coupled device camera for low-angle elastic light scattering,” Rev. Sci. Instrum. |

24. | A. K. Jain, |

25. | B.T. Draine and P.J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A |

26. | A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener, and G. von Freymann, “Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths,” Nat. Materials |

27. | I. Staude, M. Thiel, S. Essig, C. Wolff, K. Busch, G. von Freymann, and M. Wegener, “Fabrication and characterization of silicon woodpile photonic crystals with a complete bandgap at telecom wavelengths,” Opt. Lett. |

**OCIS Codes**

(030.6600) Coherence and statistical optics : Statistical optics

(290.4210) Scattering : Multiple scattering

(160.5293) Materials : Photonic bandgap materials

(160.5298) Materials : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: November 23, 2012

Revised Manuscript: December 20, 2012

Manuscript Accepted: December 21, 2012

Published: January 9, 2013

**Citation**

Jakub Haberko and Frank Scheffold, "Fabrication of mesoscale polymeric templates for three-dimensional disordered photonic materials," Opt. Express **21**, 1057-1065 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-1-1057

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### References

- J. D. Joannopoulos, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University Press, 2008).
- W. H. Wang, C. Dong, and C. H. Shek, “Bulk metallic glasses,” Mat Sci Eng R44, 45–89 (2004). [CrossRef]
- E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys Rev Lett58, 2059–2062 (1987). [CrossRef] [PubMed]
- S. John, “Localization of photons in certain disordered dielectric superlattices,” Phys Rev Lett58, 2486–2489 (1987). [CrossRef] [PubMed]
- C. Lopez, “Materials aspects of photonic crystals,” Adv. Mater.15, 1679–1704 (2003). [CrossRef]
- M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nature Materials3, 444–447 (2004). [CrossRef] [PubMed]
- M. Florescu, S. Torquato, and P. J. Steinhardt, “Designer disordered materials with large, complete photonic band gaps,” Proc. Natl. Acad. Sci.106, 20658–20663 (2009). [CrossRef] [PubMed]
- S. F. Liew, J.-K. Yang, H. Noh, C. F. Schreck, E. R. Dufresne, C. S. OHern, and H. Cao, “Photonic band gaps in three-dimensional network structures with short-range order,” Phys. Rev. A.84, 063818 (2011). [CrossRef]
- M. Florescu, S. Torquato, and P. J. Steinhardt, “Complete band gaps in two-dimensional photonic quasicrystals,” Phys. Rev. B80, 155112 (2009). [CrossRef]
- W. N. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, “Experimental measurement of the photonic properties of icosahedral quasicrystals,” Nature436, 993–996 (2005). [CrossRef] [PubMed]
- P. J. Steinhardt and D. P. Divincenzo, Quasicrystals : The State of the Art (World Scientific, 1999).
- C. J. Jin, X. D. Meng, B. Y. Cheng, Z. L. Li, and D. Z. Zhang, “Photonic gap in amorphous photonic materials,” Phys. Rev. B.63, 195107 (2001). [CrossRef]
- M. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and S. Mordechai, “Amorphous photonic lattices: band gaps, effective mass, and suppressed transport,” Phys. Rev. Lett.106, 193904 (2011) [CrossRef] [PubMed]
- L. F. Rojas-Ochoa, J. M. Mendez-Alcaraz, P. Schurtenberger, J. J. Saenz, and F. Scheffold, “Photonic properties of strongly correlated colloidal liquids,” Phys. Rev. Lett.93, 073903 (2004). [CrossRef] [PubMed]
- M. Reufer, L. F. Rojas-Ochoa, P. Schurtenberger, J. J. Saenz, and F. Scheffold, “Transport of light in amorphous photonic materials,” Appl. Phys. Lett.91, 171904 (2007). [CrossRef]
- J. F. Galisteo-Lopez, M. Ibisate, R. Sapienza, L. S. Froufe-Prez, A. Blanco, and C. Lopez, “Self-assembled photonic structures,” Adv Mater23, 30–69 (2011). [CrossRef]
- H. Noh, J.-K. Yang, S. F. Liew, M. J. Rooks, G. S. Solomon, and H. Cao, “Control of lasing in biomimetic structures with short-range order,” Phys Rev Lett106, 183901 (2011). [CrossRef] [PubMed]
- C. E. Zachary, Y. Jiao, and S. Torquato, “Hyperuniform long-range correlations are a signature of disordered jammed hard-particle packings,” Phys. Rev. Lett.106, 178001 (2011). [CrossRef] [PubMed]
- W. N. Man, M. Florescu, K. Matsuyama, P. Yadak, S. Torquato, P. J. Steinhardt, and P. Chaikin, “Experimental observation of photonic bandgaps in hyperuniform disordered material” in Proceedings of the Conference on Lasers and Electro-Optics (Cleo) and Quantum Electronics and Laser Science Conference (Qels) (2010).
- C. Song, P. Wang, and H. A. Makse, “A phase diagram for jammed matter,” Nature453, 629–632 (2008). Data taken from Hernán Makse’s web page, City College of New York (USA), http://lev.ccny.cuny.edu/hmakse/ . [CrossRef]
- A. Donev, S. Torquato, and F. H. Stillinger, “Pair correlation function characteristics of nearly jammed disordered and ordered hard-sphere packings,” Phys. Rev. E71, 011105 (2005). [CrossRef]
- J.-L. Barrat and J.-P. Hansen, Basic Concepts for Simple and Complex Liquids (Cambridge University Press, 2003). [CrossRef]
- F. Ferri, “Use of a charge coupled device camera for low-angle elastic light scattering,” Rev. Sci. Instrum.68, 2265–2274 (1997). [CrossRef]
- A. K. Jain, Fundamentals of Digital Image Processing (Prentice Hall, 1989).
- B.T. Draine and P.J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A1, 1491–1499 (1994). [CrossRef]
- A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener, and G. von Freymann, “Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths,” Nat. Materials5, 942–945 (2006). [CrossRef]
- I. Staude, M. Thiel, S. Essig, C. Wolff, K. Busch, G. von Freymann, and M. Wegener, “Fabrication and characterization of silicon woodpile photonic crystals with a complete bandgap at telecom wavelengths,” Opt. Lett.35, 1094–1096 (2010). [CrossRef] [PubMed]

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